Probability-Theory-Problems.pdf

8/19/2019 Probability-Theory-Problems.pdf

1/11


2/11

7. An electronic assembly consists of two subsystems, say A and B. Suppose wehave the following information: P (B fails) = 0.5, P (A and B fail) = 0.3 andP (A fails, but B doesnt fail) = 0.1. Find the probability that B fails, given

that A fails.8. Let (Ω, K, P ) a probability field and A, B ∈ K such that P (A ∩ B ) = 0.01,

P (A ∩ B) = 0.03, P (A ∩ B) = 0.05. Calculate P (A) + P (A ∪ B) + P (B/A).

9. Let (Ω, K, P ) a probability field and A, B,C ∈ K such that P (A) = 13

, P (B) = 14

,P (C ) = 15 , P (A ∩ B) =

16

, P (A ∩ C ) = 18 , P (B ∩ C ) = 110

and P (A ∩ B ∩ C ) = 112 .

Calculate P (C/A ∩ B).

10. Consider (Ω, K, P ) a probability field and A1, A2, A3 events in K such that A1∩A3=Ø and P (A1 ∩ A2) = P (A2 ∩ A3) = 0.2, P (A1 ∪ A2 ∪ A3) = 0.8, P (A1) =P (A3) = 0.3. Find P (A2 ∩ A1 ∪ A3) and P (A2/A3). Are A1, A2 independent

events? Justify your answer. Are the events A2 and A3 mutually exclusive?Why?

11. The same problem as the previous, with: P (A1 ∩ A2) = P (A2 ∩ A3) = 0.1,P (A1 ∪ A2 ∪ A3) = 0.9, P (A1) = P (A3) = 0.2.

12. Let {Ω, K,P } be a probability field and A, B ∈ K, with 0 < P (A) < 1, 0 <P (B) < 1. Find P (A) and P (B) if P (A/B) = 2

5, P (A/B) = 1

10 and P (B/A) = 3

5.

13. How many times one needs to roll a die, to get the face ’6’ at least once with aprobability greater than 0.7?

14. The letters m, m, a, a, e, t, h, h, o, r, r are written each one on separate small sheetsof paper. Six sheets of paper are randomly extracted, one by one. Find theprobability to obtain the word mother.

15. The Sebastien Cabinet Company, manufacturer of high-priced wood cabinets,employs 50 skilled cabinetmakers, 26 of whom have received formal trade schooland apprenticeship training and 24 of whom are self-taught. The productionmanager categorized these craftsman according to type of training received andaccording to quality of work, 1 being the best possible rating. The bivariatefrequency distribution is shown in the table below.

Tabela 1: Quality Rating of WorkType of training 1 (B1) 2 (B2) 3 (B3) 4 (B4) TotalTrade school and apprenticeship (A1) 6 11 7 2 26Self − taugh (A2) 3 6 10 5 24Total 9 17 17 7 50

What is the probability that a cabinet maker picked at random will

2


3/11

(a) Be self-taught?

(b) Do work classified as quality-level 1?

(c) Do work classified as quality-level 1 given that he or she is self-taught?

(d) Do work classified as quality-level 3 or 4 given that he or she is self-taught?

(e) Be self-taught given he or she achieved a quality level rating of 2?

(f) Be self-taught or do work classified as quality-level 3?

(g) Do work classified as quality-level 1 and be self-taught?

(h) Be formally trained and do work classified as either quality-level 2 or quality-level 3?

(i) Do work classified as quality-level 3 given that he or she is either formallytrained or self-taught?

16. Refer to table above.

(a) Does P (A1/B1) = P (B1/A1) ?

(b) Does P (A1 ∩ B1) taken directly from the table equal P (A1)P (B1) ? Whatdoes your answer convey about the independence, or lack of it, of events A1and B1 ?

(c) Are events A1 and B2 independent? Explain.

(d) Are events A1 and B3 independent? Explain.

(e) Are events A2 and B2 independent? Explain.

17. In a particular town, 20 percent of people buy the morning newspaper, 30 percentbuy the evening newspaper, and 10 percent buy both. What is the probabilitythat a person from this town buys at least one of the two newspapers?

18. A recent customer taste test of the top three soft drinks yielded the followingresults:

Category P ref erence (percentage)Coca Cola 25Pepsi 257UP 8

Coca Cola and Pepsi 12Coca Cola and 7UP 5Pepsi and 7UP 4Coca Cola and Pepsi and 7UP 3

What is the probability of a participant selected at random preferring Coca Colaor Pepsi or 7UP?

19. Consider an experiment of flipping a fair coin three times.

3


4/11


5/11

26. 6% of Type A spark plugs are defective, 4% of Type B spark plugs are defective,and 2% of Type C spark plugs are defective. A spark plug is selected at randomfrom a batch of spark plugs containing 50 Type A plugs, 30 Type B plugs, and 20

Type C plugs. The selected plug is found to be defective. What is the probabilitythat the selected plug was of Type A?

27. When Don plays tennis, 65% of his first serves are correct. If the first serve iscorrect, his chances of winning the point are 90%. If his first serve is not correct,Don is allowed a second serve, and of these, 80% are good. If the second serve isa good one, his chances of winning the point are 60%. If neither serve is correct,Don loses the point.

(a) Find the probability that Don loses the point.

(b) Find the conditional probability that Don’s first serve was correct, given

that he lost the point.28. The chances of the Los Angeles Lakers winning at home are 70%, whereas the

chances of winning on the road are 50%. The Lakers are scheduled to play twohome games followed by two road games in the coming weeks.

(a) What is the probability of the Lakers winning all four games?

(b) What is the probability of the Lakers winning three out of four games?

(c) What are the chances of the Lakers losing all four games?

29. 25 percent of students at a large university smoke cigarettes. Consider a group

of 12 randomly selected students.(a) What is the probability that exactly 5 will smoke?

(b) What is the probability that 5 or less will smoke?

(c) What is the probability that either 4 or 5 will smoke?

30. 15 percent of students on a course are not satisfied with the textbook used.Consider a group of 10 randomly selected students.

(a) What is the probability that exactly 3 will not be satisfied with the tex-tbook?

(b) What is the probability that 3 or more will not be satisfied?(c) What is the probability that 3 or less will not be satisfied?

31. The table below summarizes 60 responses to a survey question: ’Do you favor ano-smoking rule on airplanes?’

Male FemaleYes 24 14No 16 6

5


6/11

(a) What is the probability that a randomly selected person answers ’yes’?

(b) What is the probability of selecting someone from the group who answered’yes’ given that the selected person is a male?

(c) Are events of being a male and answering ’yes’ independent?

32. Three boxes contain black and white balls, as follows. Box 1:3 white and 5 black,Box 2: 5 white and 3 black, Box 3: 2 white and 3 black. One ball is extractedfrom each box, without replacement.

(a) What is the probability to obtain one white ball?

(b) What is the probability to obtain only black balls?

(c) Suppose that after extracting one ball from each box, the three balls arereintroduced in their correspondent boxes and this operation is repeated

5 times. What is the probability to obtain exactly 3 times the followingcombination (2 white balls, 1 black ball)?

33. Three boxes contain black and white balls, as follows. Box 1: 3 white and 4black, Box 2: 5 white and 10 black, Box 3: 7 white and 2 black. One ball isextracted 2 times with replacement from the first box, while from the second andthird one are extracted two balls, without replacement. What is the probabilityto obtain the following combination: (1 white ball, 1 black ball) either from thefirst box or from the second and third boxes, or both?

34. Consider 3 boxes with white and black balls as follows: B1:5 white and 5 black,

B2: 4 white and 6 black, B3: 4 white and 5 black. We extract, with replacement,5 balls from each box. Find the probability to obtain the combination (2 whiteand 3 black) from 2 boxes and from the third, any other combination.

35. Consider 3 boxes with white and black balls as follows: B1: 5 white and 5 black,B2: 4 white and 6 black, B3: 4 white and 5 black. We extract, without replace-ment, 5 balls from each box. Find the probability to obtain the combination (2white and 3 black) from 2 boxes and from the third, any other combination.

2 Random Variables

36. Consider a box containing 6 white balls and 4 black balls. Three balls are ex-tracted at random and let X be the random variable representing the numberof white balls extracted. Write the distribution table of the random variable X ,assuming that the extractions are done with replacement.

37. Consider a box containing 3 while balls and 2 black balls. Three balls are ex-tracted at random and let X be the random variable representing the number

6


7/11

of white balls extracted. Write the distribution table of random variable X ,assuming that the extractions are done without replacement.

38. Consider the function F : R → R, with

F (x) =

a if x ≤ 0kx2 if 0 < x ≤ 1

b if x > 1, a,b,k ∈ R

(a) Determine a, b, k ∈ R such that F is a cumulative distribution function.

(b) Determine P (14 ≤ X ≤ 34).

39. Consider the following two discrete, independent random variables:

X : −1 0 1a + 16 b + 13 13

, Y : −1 0 113 2a − b 12a2

(a) Write the distribution table of the random variable 2XY .

(b) Determine k ∈ Z such as P (X + Y = k) > 29 .

40. Consider X, Y independent random variables,

X :

−1 0 13 p 2 p 5 p

, Y :

0 1 2 3

3q 2q q 4q

and F XY the cumulative distribution function of random variable X Y . Compute

F XY (3) − F XY (1).

41. Let

X :

2 3 50.2 0.3 0.5

, Y :

1 4 60.6 0.2 0.2

two independent discrete r.v. on the same probability field. Determine the dis-tribution tables for X + Y , X − Y , XY and calculate E (X ), E (Y ), V ar(X ),V ar(3X − 5), V ar(7XY ).

42. Calculate the expectation and variance of

(a) the Binomial random variable with parameters n and p;

(b) the Geometric random variable with parameter p;

(c) the Poisson random variable with parameter λ.

43. Let (X, Y ) be the bivariate discrete random vector with distribution table

(a) Determine the marginal distributions of the random variables X, Y . Are ther.v. X, Y independent?

7


8/11

X Y -2 0 2

2 0.2 0.25 0.153 0.1 0.25 0.05

(b) Determine the distribution of the random variable X +Y and the conditionaldistribution of (X + Y |X = 3).

(c) Compute F (X,Y )(52 ,

12) where F (X,Y )(x, y) is the bivariate c.d.f of the r.v

(X, Y ).

44. Consider the independent r.v. X and Y with E (X ) = −1, E (Y ) = 1, V ar(X ) =V ar(Y ) = σ2. Determine V ar(XY ).

45. Consider the discrete r.v. X ,

X :

n

pn

n∈N ∗

with pn = e−λ(1 − e−λ)n−1. Calculate E (X ) and V ar(X ).

46. Two fair dice are rolled. Let the random variable X be the smaller of the twoscores if the dice show different faces, or the common score if the dice show thesame face. Write the distribution table of X and calculate E (X ).

47. Consider two discrete random variables X, Y with the following distribution ta-bles:

X :

−1 112

12

, Y :

−1 223

13

and let P (X = −1, Y = −1) = λ, where λ is a real parameter.

(a) Determine the distribution table of the random vector Z = (X, Y ) (depen-ding on λ);

(b) Find the correlation coefficient of X and Y ;

(c) Find the value of λ for which X and Y are uncorrelated. For this value of λ, are the random variables X and Y independent?

48. Consider the discrete random variable (X,Y) with the distribution table

X Y -2 0 1

-1 0.1 0.1 0.3-2 0.2 0.2 0.1

Calculate E (X ), E (Y ), V ar(X + Y ) and ρ(X, Y ). Are the random variables X and Y independent? Justify your answer.

8


9/11

49. Consider the discrete random variables (X, Y ) with distribution table below.Calculate E (X ), E (Y ), V ar(X + Y ) and ρ(X, Y ). Are the random variables X and Y independent? Justify your answer.

X Y -2 1 5

-1 0.125 0.25 0.1251 0.125 0.25 0.125

50. Consider the discrete bivariate random variable Z = (X, Y ) given by the followingtable. Find x,y,a,b, c, d such that E (X ) = 13 , E (Y ) =

14 and compute F (X,Y )(0, 2)

and ρ(X, Y ).

X Y 0 y

x a 16

b1 712 c

23

34 d

51. Consider a box containing 1 white, 2 black and 3 blue balls. We extract, wi-thout replacement, two balls and denote by X and Y the random variablesrepresenting the number of white and, respectively black balls obtained fromthe box. Compute ρ(X, Y ), V ar(2X + 3Y ), P (0.5 ≤ min{X, Y } < 1.5) andP (−1

2

≤ min{X, Y } < 5

2

).

52. Consider a box containing 1 white and 2 black balls. We extract, with replace-ment, two balls and denote by X the random variable representing the number of white balls obtained. Determine the distribution of random variable X , computeρ(X, X 2) and the probability that the extracted balls have the same color.

53. Consider three boxes containing white and black balls as follows: U1 : 1 whiteand 2 black, U2 : 2 white and 3 black and U3 : 1 white and 2 black balls. Weextract without replacement, 2 balls from U1 and one ball from U2 and U3 (onefrom each). Denote by X the random variable representing the number of whiteballs obtained from U1 and by Y , the random variable representing the number

of white balls obtained from U2 and U3. Find the distribution of Z = (X, Y )and ρ(X, Y ).

54. Consider a box containing 25% white, 50% black and 25% blue balls. We extractwith replacement 2 balls. Denote by X and Y the random variables representingthe number of white and, respectively black balls obtained. Find the distributionof X Y and ρ(X, Y ).

9


10/11

55. Consider two boxes, B1, B2 containing white and black balls as follows: B1: 1white and 2 black, B2: 2 white and 2 black. We extract one ball from eachbox and denote by X the random variable representing the number of white balls

obtained. Determine the distribution of X , compute ρ(X, X

2

) and the probabilitythat the extracted balls have the same color.

56. Let X be a random variable with the probability density function (pdf)

f (x) = cxe−x

3 , x > 0.

Determine the parameter c, the cumulative distribution function F , E (X ) andP (X ≤ 2/X > 0).

57. Let X a random variable having the probability density function f (x) = e−2|x|,x ∈ R. Calculate P (|X | < n).

58. Let X and Y two random variables defined on the same probability field.

(a) If X and Y are independent random variables with V ar(X ) = 3, V ar(Y ) =7, find V ar(2X − 3Y ).

(b) Repeat your calculations dropping the assumptions of independence andusing instead the information that cov(X, Y ) = 1.

59. Suppose N is a discrete r.v. with Geometric distribution: N ∼

n

p · q n−1

,

where p, q ≥ 0, p + q = 1, n ∈ N∗. Show that P (N > t + n|N > t) = P (N > n),

∀n > 0, t > 0.

60. A continuous random variable X has pdf f (x) =

αx(1 − x) if 0 ≤ x ≤ 1;0 otherwise.

(a) Find α, E (X ) and V ar(X ).

(b) Find the cdf F (·) of X and calculate the probability P (1/2 < X ≤ 3/4).

61. Let X be a discrete r.v. with the following probability mass function:

P (X = n) = αn

(1+α)1+n, n ∈ N, α > 0. Find E (X ) and V ar(X ).

62. Let X ∼ Exp(λ) with f (x) =

λe−λx

if x > 0;0 otherwise.

Find the cdf F (·) of X and show that P (X > s +t|X > t) = P (X > s), ∀s, t > 0.

63. Suppose N is a discrete r.v. with Geometric distribution: N ∼

n

p · q n−1

,

where p, q ≥ 0, p + q = 1, n ∈ N∗. What is the probability that N is an evennumber?

10


11/11

64. A uniform random variable X has pdf f (x) =

14 if −2 ≤ x ≤ 2;0 otherwise.

(a) Find the cdf F (·) of X , E (X ) and V ar(X ).

(b) Find the pdf of Y = eX .

65. The quality of an electronic device is given by two characteristics U and V havingthe following probabilistic models: U = 2X + 3Y and V = 4X − Y where X ,Y are independent random variables, X ∼ N (3, 2), Y ∼ Beta(10, 0.9). Find thecorrelation coefficient ρ(U, V ), using the formulas for the moments of Normal andBeta distributions.

11

Probability-Theory-Problems.pdf

Documents

Transcript of Probability-Theory-Problems.pdf